The most conventional mechanism for gravitational waves (gw) production during inflation is the amplification of vacuum metric fluctuations. In this case the gw production can be uniquely related to the inflationary expansion rate H. For example, a gw detection close to the present experimental limit (tensor-to-scalar ratio r ~0.1) would indicate an inflationary expansion rate close to 10^14 GeV}. This conclusion, however, would be invalid if the observed gw originated from a different source. We construct and study one of the possible covariant formulations of the mechanism suggested in Biagetti, Fasiello and Riotto (2013) where a spectator field with a sound speed cs << 1 acts as a source for gw during inflation. In our formulation the spectator field is described by a so-called P(X) Lagrangian and a non-minimal coupling to gravity. This field interacts only gravitationally with the inflaton, which has a standard action. We compute the amount of scalar and tensor density fluctuations produced by the spectator field and find that, in our realization, r is not enhanced with respect to the standard result but it is strongly sensitive to cs thus breaking the direct r -> H connection.